2. Analysis of wave equation

Consider the propagation of a circularly polarized laser pulse in a preformed plasma channel having a parabolic density profile of the form $n(r)=n_{0}+{\rm\Delta}n\,r^{2}/r_{ch}^{2}$ , where $n_{0}$ is the ambient plasma density, $n(r)$ is the density of the plasma at distance $r$ from the axis of the channel, and ${\rm\Delta}n$ and $r_{ch}$ are the channel depth and radius, respectively. The transverse radiation field in terms of the vector potential is given by

where $A(r,z,t)$ is the slowly varying field envelope, $k$ and ${\it\omega}$ are the wavenumber and frequency, respectively, and ${\it\sigma}$ is a sign factor that is $+1$ for a left-turn pulse and $-1$ for a right-turn pulse. The longitudinal component of the radiation field has been neglected (Cicchitelli, Hora & Postle Reference Cicchitelli, Hora and Postle1990). Using Coulomb gauge $(\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{A}=0)$ and obtaining the relativistic electron velocity from the equation of plasma motion, the propagation of the radiation field in the presence of nonlinear source terms can be described by

where $k_{p0}={\it\omega}_{p0}/c=(4{\rm\pi}n_{0}e^{2}/mc^{2})^{1/2}$ is the plasma wavenumber, ${\it\Omega}_{i}$ is a magnetic factor that is equal to ${\it\omega}/({\it\omega}+{\it\omega}_{c})$ for right-turn and to ${\it\omega}/({\it\omega}-{\it\omega}_{c})$ for left-turn pulses, ${\it\omega}_{c}$ is the electron cyclotron frequency in an external magnetic field and $\boldsymbol{a}=e\boldsymbol{A}/mc^{2}$ is the normalized vector potential, having a Gaussian profile of the form $|\boldsymbol{a}|^{2}=a_{0}^{2}(r_{0}^{2}/r_{s}^{2})\exp (-2r^{2}/r_{s}^{2})$ , with $a_{0}$ , $r_{0}$ and $r_{s}$ being the peak amplitude, beam waist and spot size, respectively. In the absence of an external field, the magnetic factor will be ‘1’ and the behaviour of a circularly polarized pulse is of the same form as that of a linearly polarized pulse. This has been studied by Pallavi et al. (Reference Pallavi, Navina, Ajay and Gaurav2004) for a linear polarization pulse in the absence of an external field. If $r^{2}/r_{s}^{2}\ll 1$ is assumed, only terms up to order $r^{2}/r_{s}^{2}$ may be retained in the exponential expansion of the intensity profile. Introducing the independent variables $z$ and ${\it\zeta}=z-{\it\beta}_{g}ct$ ( ${\it\beta}_{g}=v_{g}/c=ck/{\it\omega}$ is the normalized group velocity of the pulse) and substituting (2.1) into (2.2) gives

where ${\it\beta}_{p}=({\it\omega}/ck)(1+k_{p0}^{2}{\it\Omega}_{i}/k^{2}+4/k^{2}r_{0}^{2}-k_{p0}^{2}{\it\Omega}_{i}a_{0}^{2}/4k^{2})^{1/2}$ is the normalized phase velocity, which comprises the Gaussian amplitude of the laser beam, nonlinear relativistic effects and the magnetic field.

In the approximation of short laser pulses ( $L<{\it\lambda}_{p}$ ) propagating in a plasma channel, the last term on the left-hand side of (2.3) will be neglected in comparison to the third and fourth terms in the limit $L\ll 2Z_{R0}$ and $L<r_{0}$ , respectively, where $Z_{R0}=kr_{0}^{2}/2$ . These terms represent non-paraxial propagation and lead to a short pulse and group velocity dispersion effects. To solve this second-order differential equation, the Fourier transform method that appears in Yariv’s (Reference Yariv1989) book is used. The Fourier transform of (2.3) with respect to ${\it\zeta}$ (where $a_{k}=(2{\rm\pi})^{-1/2}\int _{-\infty }^{\infty }\text{d}{\it\zeta}\exp (-\text{i}{\it\delta}k{\it\zeta})a({\it\zeta})$ is the Fourier transform of $a$ and ${\it\delta}k\ll k$ is the spread in wavenumber due to a finite pulse length) gives

The left-hand side of (2.4) is similar to a paraxial wave equation and has a standard Gaussian beam solution (in $k$ space) of the form

where $b_{k}(k,z)$ , ${\it\theta}_{k}(k,z)$ , ${\it\alpha}_{k}(k,z)$ and $r_{s}(k,z)$ are the amplitude, phase shift, curvature and spot size, respectively. Substituting (2.5) into (2.4) and equating coefficients of $r^{0}$ , the imaginary and real parts give

and

where $k_{0}=k+{\it\delta}k$ . Similarly, equating coefficients of $r^{2}$ , the imaginary and real parts give

and

where $r_{e}=e^{2}/mc^{2}$ is the classical electron radius and $r_{m}=(r_{ch}^{2}/{\rm\pi}r_{e}{\rm\Delta}n)^{1/4}=(4n_{0}r_{ch}^{2}/K_{p0}^{2}{\it\Omega}_{i}{\rm\Delta}n)^{1/4}$ is the matched spot radius. Equation (2.7b ) describes the evolution of the laser spot. The first, second and third terms on the right-hand side of (2.7b ) represent vacuum diffraction, relativistic self-focusing and channel focusing, respectively. The last (defocusing) term arises due to coupling of the relativistic nonlinearity with the plasma channel. The last three terms are dependent on the magnetic field, and hence the magnetic field effects on the laser spot behaviour are explicit.

3. Evolution of spot size

The solution to (2.7b ), assuming the initial conditions $\text{d}r_{s}/\text{d}z=0$ and $r_{s}=r_{i}$ at $z=0$ , is given by

Equation (3.1) shows that the laser spot undergoes betatron oscillations with wavenumber $k_{{\it\beta}m}$ ( $=2/Z_{Rm}$ , with $Z_{Rm}=k_{0}r_{m}^{2}/2$ ). The wavenumber is seen to be independent of relativistic nonlinearities and to be dependent on the magnetic field. Changing the magnetic field causes a change in the focusing distance of the laser pulse. As the magnetic factors of the right-turn and left-turn pulses are different, the behaviour of these pulses will be different. Also, in the absence of a magnetic field, a circular pulse behaves similarly to a linear pulse. However, the amplitude of oscillation varies from $r_{s}=r_{i}$ to $r_{s}=(r_{m}^{2}/r_{i})(1-k_{p0}^{2}{\it\Omega}_{i}a_{0}^{2}r_{0}^{2}/8+a_{0}^{2}r_{0}^{2}r_{i}^{2}/4r_{m}^{4})^{1/2}$ with an average value $\langle r_{s}\rangle =(r_{i}/2)[1+(r_{m}^{2}/r_{i}^{2})(1-k_{p0}^{2}{\it\Omega}_{i}a_{0}^{2}r_{0}^{2}/8+a_{0}^{2}r_{0}^{2}r_{i}^{2}/4r_{m}^{4})^{1/2}]$ that is a function of relativistic nonlinearity and magnetic field.

In order to analyse the effect of nonlinearity on the propagation of a broad laser pulse ( ${\it\lambda}_{p}=15~{\rm\mu}\text{m}$ and $r_{0}=10~{\rm\mu}\text{m}$ ), the evolution of its spot, as given by (3.1), for $r_{i}=r_{0}$ , is plotted against the normalized propagation distance $z/Z_{Rm}$ in figure 1, for a laser beam of wavelength $1~{\rm\mu}\text{m}$ and intensity $1.4\times 10^{17}~\text{W}~\text{cm}^{-2}$ (or $4.1\times 10^{17}~\text{W}~\text{cm}^{-2}$ ), $a_{0}^{2}=0.1$ (or 0.3). For these two values of $a_{0}^{2}$ , nonlinear propagation of the laser spot is shown by dotted and dashed curves, respectively. The latter curve shows a phase reversal of ${\rm\pi}/2$ with respect to the former. This phase reversal occurs since in this case the channel depth exceeds the nonlinear critical channel depth. Also, it may be noted that in the presence of relativistic nonlinearities the amplitude of betatron oscillation is reduced in comparison to the linear case (solid curve), since, for a broad pulse, the relativistic self-focusing effects dominate over the defocusing effect of channel coupling. The reduction in the amplitude of the spot size is therefore due to the relativistic self-focusing effect, which tends to focus the laser spot close to the axis. On the contrary, for a narrow pulse, the amplitude of betatron oscillation would increase due to the channel-coupling defocusing effect.

Figure 1.

Normalized spot size ( $r_{s}/r_{0}$ ) against normalized propagation distance ( $z/Z_{Rm}$ ) inside the magnetic field ${\it\omega}_{c}/{\it\omega}=0.1$ in the absence of nonlinearities (solid curve), for $a_{0}^{2}=0.1$ (dotted curve), for $a_{0}^{2}=0.2$ (dashed curve) and for $a_{0}^{2}=0.3$ (blue dotted curve) with ${\it\lambda}_{p}=15~{\rm\mu}\text{m}$ , ${\rm\Delta}n=2\times 10^{18}~\text{cm}^{-3}$ , $r_{ch}=20~{\rm\mu}\text{m}$ and $r_{0}=10~{\rm\mu}\text{m}$ : (a) right-turn pulse; (b) left-turn pulse.

As specified from figure 1, inside the magnetic field, a right-turn pulse is more defocused than a left-turn pulse. This opposite behaviour is observed even in the non-relativistic case. In figure 2, laser spot oscillation is plotted in the absence of an external magnetic field; the curves for right-turn and left-turn pulses overlap and are uniform with the linear polarized pulse. Under these conditions, in the absence of an external magnetic field, these pulses are matched with a plasma channel of intensity $a_{0}^{2}=0.2675$ (self-trapping phenomena). In the presence of an external field ${\it\omega}_{c}/{\it\omega}=0.1$ , the right-turn pulse of $a_{0}^{2}=0.3155$ and left-turn pulse of $a_{0}^{2}=0.2195$ are matched.

Figure 2.

Normalized spot size ( $r_{s}/r_{0}$ ) against normalized propagation distance ( $z/Z_{Rm}$ ) in the absence of an external magnetic field, for the linear case (solid curve), for $a_{0}^{2}=0.1$ (dotted curve), for $a_{0}^{2}=0.2$ (dashed curve) and for $a_{0}^{2}=0.3$ (blue dotted curve) with ${\it\lambda}_{p}=15~{\rm\mu}\text{m}$ , ${\rm\Delta}n=2\times 10^{18}~\text{cm}^{-3}$ , $r_{ch}=20~{\rm\mu}\text{m}$ and $r_{0}=10~{\rm\mu}\text{m}$ .

Comparing the right-turn and left-turn curves in figure 1, it is shown that the wavelength of oscillations in these curves is not uniform. The wavelength of the left-turn pulse is smaller. Figure 3 shows that the left-turn pulse is shorter and also more focused than the right-turn pulse. This shows that, in the absence of a magnetic field, the right-turn and left-turn pulses have the same behaviour. But in the presence of a magnetic field, the left-turn pulse is more focused than the other one. With increase of the magnetic field, this effect is greater.

Figure 3.

Normalized spot size ( $r_{s}/r_{0}$ ) against normalized propagation distance ( $z/Z_{Rm}$ ) in the presence/absence of an external magnetic field for right-turn pulse ( $R$ ) and left-turn pulse ( $L$ ), for $a_{0}^{2}=0.3$ .

The effects of the other parameters on the spot size behaviour can also be shown. Increasing the channel radius causes the laser pulse to defocus. Relativistic nonlinearities cause the laser pulse to focus. The presence of an external magnetic field causes the left-turn pulse to focus and the right-turn pulse to defocus. Increasing the channel depth causes the laser beam to focus.

In a circularly polarized pulse, the wavenumber of the laser pulse $k_{{\it\beta}m}$ depends on the external field and is $k_{{\it\beta}m}=2(1+{\it\omega}_{c}/{\it\omega})^{-1/2}$ and $k_{{\it\beta}m}=2(1-{\it\omega}_{c}/{\it\omega})^{-1/2}$ for a right-turn pulse and a left-turn pulse, respectively. In the first focusing point, the amount of shift of the focus point is 0.287 for a right-turn pulse with ${\it\omega}_{c}/{\it\omega}=0.1$ to ${\it\omega}_{c}/{\it\omega}=0.3$ and is $-0.118$ for a left-turn pulse with ${\it\omega}_{c}/{\it\omega}=0.1$ to ${\it\omega}_{c}/{\it\omega}=0.17$ .

For the linear case, the condition for a laser beam to propagate with a constant spot size equal to the beam waist is $r_{i}=r_{m}=r_{0}$ and ${\rm\Delta}n={\rm\Delta}n_{c}$ . If this condition is applied to (3.1) for the nonlinear case, it reduces to

where $S=k_{p0}^{2}{\it\Omega}_{i}a_{0}^{2}r_{0}^{2}/16-a_{0}^{2}/8$ . It may be noted from (3.2) that the laser spot is no longer matched but undergoes betatron oscillations with a wavenumber $k_{{\it\beta}}$ ( $=2/Z_{R}$ , with $Z_{R}=k_{0}r_{0}^{2}/2$ ). The spot size in this case oscillates between $r_{0}$ and $r_{0}(1-S)$ with an average value of $r_{0}(1-S/2)$ . The mismatch in the laser spot arises due to relativistic self-focusing effects and defocusing due to channel coupling. If $k_{p0}{\it\Omega}_{i}r_{0}>\sqrt{2}$ (broad pulse), relativistic self-focusing dominates over channel coupling and vice versa.

If $k_{p0}{\it\Omega}_{i}r_{0}\rightarrow \sqrt{2}$ , betatron oscillations are attenuated and the spot size attains a constant value $r_{0}$ . Thus, this is an additional condition (along with ${\rm\Delta}n={\rm\Delta}n_{c}$ ) required for matching of the nonlinearly mismatched laser beam.

For nonlinear propagation of the laser pulse, the condition for matched beam propagation $r_{s}=r_{i}=r_{0}$ may be obtained from (3.1). Thus the spot size remains constant if the channel depth satisfies the condition

where ${\rm\Delta}n=r_{ch}^{2}/{\rm\pi}r_{e}r_{0}^{4}$ . Equation (3.3) represents the nonlinear critical channel depth for propagation of a laser beam for which the laser spot is independent of $z$ and equal to the beam waist. If nonlinearities are neglected, ${\rm\Delta}n_{Nc}={\rm\Delta}n_{c}$ .

A plot of normalized channel depth ${\rm\Delta}N={\rm\Delta}n_{Nc}/{\rm\Delta}n_{c}$ against $a_{0}$ for ${\it\lambda}_{p}=15~{\rm\mu}\text{m}$ and $r_{0}=10~{\rm\mu}\text{m}$ is given in figure 4. It may be seen that, owing to nonlinear effects, the critical channel depth decreases significantly with increase in $a_{0}$ . The critical channel depth is dependent on the external field, and the self-focusing/self-trapping threshold is different for right-turn, left-turn and linear pulses.

Figure 4.

Normalized channel depth ${\rm\Delta}N$ against $a_{0}$ for ${\it\lambda}_{p}=15~{\rm\mu}\text{m}$ and $r_{0}=10~{\rm\mu}\text{m}$ .

4. Evolution of laser field amplitude

The condition for the propagation of a nonlinearly matched laser pulse (spot matched with channel and propagated with fixed spot) is given by ${\rm\Delta}n={\rm\Delta}n_{Nc}$ and $r_{i}=r_{0}$ . Substituting these conditions into (3.1) gives $r_{s}=r_{0}$ . Thus, for matched beam propagation, the solutions of (2.6a ), (2.6b ) and (2.7a ) (for initial conditions ${\it\theta}={\it\alpha}=0$ at $z=0$ ) are given by $b_{k}=b_{k0}$ , ${\it\theta}=-{\it\delta}k^{2}(1-{\it\beta}_{g}^{2})z/2k_{0}$ and ${\it\alpha}=0$ , respectively.

The wave amplitude (2.5), for a matched pulse, therefore reduces to

The $z$ dependence of $a_{k}$ in (4.1) characterizes the nonlinear group velocity dispersion effect. Retaining terms up to second order in ${\it\delta}k$ , the inverse Fourier transform of (4.1) is given by

where $R=z/Z_{D}$ , with $Z_{D}=({\rm\pi}L/{\it\lambda})^{2}Z_{R0}(1+k_{p0}^{2}{\it\Omega}_{i}r_{0}^{2}/4-k_{p0}^{2}{\it\Omega}_{i}a_{0}^{2}r_{0}^{2}/16)^{-1}$ the characteristic length for dispersive spreading, which is a function of relativistic nonlinearity and external magnetic field. Dispersion increases the pulse length ( $L$ ) to an effective value $L_{e}=L(1+R^{2})^{1/2}$ . For example, at $z=20Z_{R0}$ , a linear pulse of length $L=5~{\rm\mu}\text{m}$ in the paraxial limit increases to $L=5.42~{\rm\mu}\text{m}$ when non-paraxial effects are taken into account (with $a_{0}^{2}=0.3$ , ${\it\lambda}=1~{\rm\mu}\text{m}$ , ${\it\lambda}_{p}=15~{\rm\mu}\text{m}$ and $r_{0}=10~{\rm\mu}\text{m}$ ). The same value in the presence of an external field is $L=5.49~{\rm\mu}\text{m}$ for a left-turn pulse and $L=5.36~{\rm\mu}\text{m}$ for a right-turn pulse. Further, relativistic nonlinearity contributes to the dispersive spreading and causes a reduction in the broadening of the pulse length. If the dispersion effect is not taken into account, (4.2) reduces to

which is independent of the magnetic field, and the intensity of the matched pulse is given by

5. Summary and conclusion

Non-paraxial, nonlinear propagation of a circularly polarized laser pulse, in a preformed magnetized plasma channel having a parabolic density profile, has been analysed. It is seen that the laser spot undergoes betatron oscillations in the plasma channel and remains guided inside the channel. Relativistic motion of plasma electrons cause nonlinearities and their coupling with the plasma channel affects the propagation of the laser spot. Relativistic nonlinearity causes the laser spot to focus, whereas channel coupling tends to defocus the spot. The amplitude of betatron oscillation changes due to these nonlinearities. For broad pulses ( $k_{p0}^{2}{\it\Omega}_{i}r_{0}^{2}/2>1$ ), the relativistic nonlinearity dominates and tends to reduce the amplitude of betatron oscillation as compared to the amplitude of oscillation for the linear case. The nonlinear critical channel depth ${\rm\Delta}n_{Nc}$ required to propagate a laser pulse for which the spot size remains constant and equal to the waist (matched spot size) is derived. It is seen that, if nonlinearities are neglected, the critical channel depth ${\rm\Delta}n_{Nc}$ reduces to the linear critical channel depth ${\rm\Delta}n_{c}$ . Hence, if a channel of depth ${\rm\Delta}n_{c}$ is considered, the propagating laser pulse would become mismatched due to nonlinear effects and an additional condition given by $k_{p0}{\it\Omega}_{i}r_{0}=\sqrt{2}$ would be required for beam matching.

Changing magnetic field causes changes in the focusing distance and also in the amplitude of the laser pulse. As the magnetic factors of the right-turn and left-turn pulses are different, the behaviour of these pulses will be different. Of course, in the absence of a magnetic field, a circular pulse behaves similarly to a linear pulse. The distance between consecutive focus points is increased for a right-turn pulse and decreased for a left-turn pulse. In the first focusing point, the amount of the focus point shift is 0.287 for a right-turn pulse with ${\it\omega}_{c}/{\it\omega}=0.1$ to ${\it\omega}_{c}/{\it\omega}=0.3$ and is $-0.118$ for a left-turn pulse with ${\it\omega}_{c}/{\it\omega}=0.1$ to ${\it\omega}_{c}/{\it\omega}=0.17$ .

A non-relativistic condition of self-trapping could not match the laser spot with the channel and then it needs to be modified. The new condition includes nonlinearity and external field effects. Neglecting relativistic nonlinearity provides the non-relativistic condition.

The laser field amplitude is derived for a matched pulse. It has been shown that, for non-paraxial propagation of a matched laser pulse, the group velocity dispersion effect causes broadening of the pulse length. However, relativistic nonlinearity reduces the broadening.